Bernhard riemann biography summary graphic organizer

If only I had the Theorems! Then I should find the proofs easily enough.

Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.

Biography

Early years

Riemann was born on September 17, 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of s

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Prime Obsession

1. Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics By John Derbyshire, Joseph Henry Press, 2003

2. The Riemann Hypothesis - 1859 All non-trivial zeros of the zeta function have real part one-half Hilbert’s 8th Great Problem 1900 Congress of Mathematicians.

3. Several Looks at the Riemann Zeta Function

4. Two Looks at the Riemann Zeros

5. The Golden Key - Euler 1737 Sieve of Eratosthenes 230 BCE Primes and all their multiples are eliminated from the RHS

6. What happens if we do this for every Prime to  ?

7. The Prime Number Theorem  (n) ~ N/ln(N) The probability that N is prime is: 1/ln(N) The Nth Prime Number is N  ln(N)  (n) is the prime number counting function. Gauss -1849

8. Improved Prime Number Theorem  (n) ~ Li(N) Li(n) ~ N  ln(N) This is a precise, unproven expression for  (n), given in Riemann’s 1859 paper. Proved in 1896, independently by Jacques Hadamard and Charles de la Vallee Poussin. Based on all non-trivial zeros having real part less than 1.

9. Of note, Hardy, 1914, infinitely many of the Riemann non-trivial zeros have real part 1/2. Littlewood, 1914, Li crosses  (x) infinitely many times. Bays and Hudson showed the lowest known violations in 2000 around 1.39822 * 10 31